Tamil Nadu HSE First Year Statistics cheat sheet

Statistics has two senses: statistical data (numerical facts) and statistical methods (collection, presentation, analysis, interpretation).

Stages of a statistical study: collection, organisation, presentation, analysis, interpretation - interpretation without analysis is meaningless.

Limitations: statistics studies aggregates of facts, not isolated individuals; results are true only on average and only under comparable conditions.

Primary data are collected by the investigator first-hand; secondary data are already collected by someone else and reused.

Census method: every unit of the population is enumerated; suitable when the population is small and high accuracy is needed.

Sample method: a part of the population is studied; suitable when the population is large, the study is destructive, or cost and time are limited.

Direct personal interview gives the most accurate data but is costly; mailed questionnaires are cheap but suffer low and biased response.

Classification is arranging raw data into homogeneous groups; variables are quantitative (discrete or continuous), attributes are qualitative.

A frequency distribution lists classes with their frequencies; class mark = (lower limit + upper limit) / 2 and class width = upper − lower limit.

Arithmetic mean (raw data): x̄ = (Σx)/n; for a frequency distribution x̄ = (Σfx)/Σf; for grouped data use class marks as x.

Step-deviation form: x̄ = A + h(Σfd/Σf) with d = (x − A)/h; A is an assumed mean and h the class width.

Combined mean of two groups: x̄ = (n₁x̄₁ + n₂x̄₂)/(n₁ + n₂).

Median (raw data): arrange in order; if n is odd the median is the ((n+1)/2)th value, if n is even it is the mean of the (n/2)th and (n/2+1)th values.

Median (grouped data): M = L + [(N/2 − c.f.)/f] × h, where L is the lower limit of the median class, c.f. the cumulative frequency before it, f its frequency, h the width.

Mode (grouped data): Z = L + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h, with f₁ the modal frequency and f₀, f₂ the neighbouring frequencies.

Empirical relation: Mode = 3 Median − 2 Mean (valid for moderately skewed distributions).

Geometric mean of n positive values: GM = (x₁x₂…xₙ)^(1/n), used for ratios and growth rates; harmonic mean HM = n/Σ(1/x), used for average speeds.

Mean is affected by every observation and by extreme values; median and mode are positional and resistant to outliers.

Range = Largest value − Smallest value; coefficient of range = (L − S)/(L + S). Range uses only two observations and ignores the rest.

Quartile deviation QD = (Q₃ − Q₁)/2; coefficient of QD = (Q₃ − Q₁)/(Q₃ + Q₁).

Mean deviation about A: MD = (Σf|x − A|)/Σf; it is least when A is the median.

Variance of raw data: σ² = (Σx²)/n − x̄²; for a frequency distribution σ² = (Σfx²)/Σf − x̄².

Standard deviation is the positive square root of variance; it has the same units as the data.

Coefficient of variation CV = (σ/x̄) × 100; it compares variability of two series in different units, and the series with smaller CV is more consistent.

Combined variance: σ² = [n₁(σ₁² + d₁²) + n₂(σ₂² + d₂²)]/(n₁ + n₂), where d₁ = x̄₁ − x̄ and d₂ = x̄₂ − x̄.

Variance is unchanged by adding a constant to every observation and is multiplied by k² when every observation is multiplied by k.

For a symmetric distribution mean = median = mode; dispersion measures are never negative and equal zero only when all observations are equal.

Skewness measures asymmetry; kurtosis measures peakedness or flatness relative to the normal curve.

Pearson's coefficient of skewness: SK = (Mean − Mode)/σ, or SK = 3(Mean − Median)/σ when the mode is ill-defined.

Bowley's coefficient: SK = (Q₃ + Q₁ − 2Q₂)/(Q₃ − Q₁); it lies between −1 and +1.

Positive skew (right tail) gives Mean > Median > Mode; negative skew (left tail) gives Mean < Median < Mode.

For a symmetric distribution SK = 0 and mean = median = mode.

Moment about the mean: μᵣ = Σf(x − x̄)ʳ/Σf; μ₁ = 0 and μ₂ = σ².

β₁ = μ₃²/μ₂³ and γ₁ = ±√β₁ give the direction and degree of skewness.

β₂ = μ₄/μ₂² and excess kurtosis γ₂ = β₂ − 3; β₂ = 3 for the mesokurtic normal curve.

β₂ > 3 is leptokurtic (peaked), β₂ < 3 is platykurtic (flat); for a normal curve γ₂ = 0.

Correlation measures the strength and direction of the linear relation between two variables; it does not imply causation.

Karl Pearson's coefficient: r = Σ(x − x̄)(y − ȳ)/[√Σ(x − x̄)² √Σ(y − ȳ)²]; it lies between −1 and +1.

Shortcut form: r = [nΣxy − ΣxΣy] / √{[nΣx² − (Σx)²][nΣy² − (Σy)²]}.

r = +1 is perfect positive, r = −1 perfect negative, r = 0 no linear correlation; r is independent of change of origin and scale.

Spearman's rank correlation: ρ = 1 − 6Σd²/[n(n² − 1)], where d is the difference in ranks; for repeated ranks use the correction Σ(t³ − t)/12.

Regression line of y on x: y − ȳ = b_yx(x − x̄) with b_yx = Σ(x − x̄)(y − ȳ)/Σ(x − x̄)² = r(σ_y/σ_x).

Regression line of x on y: x − x̄ = b_xy(y − ȳ) with b_xy = r(σ_x/σ_y); the two lines pass through (x̄, ȳ).

Relation between coefficients: r = ±√(b_yx × b_xy), the sign being that of r; since |r| ≤ 1, b_yx × b_xy ≤ 1.

If σ_x = σ_y then b_yx = b_xy = r; the two regression lines coincide only when r = ±1.

Classical definition: P(A) = number of favourable outcomes / total number of equally likely outcomes.

Axioms: 0 ≤ P(A) ≤ 1, P(S) = 1, and P(A ∪ B) = P(A) + P(B) when A and B are mutually exclusive.

Addition theorem: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); P(A′) = 1 − P(A).

Conditional probability: P(A|B) = P(A ∩ B)/P(B), defined only when P(B) > 0.

Multiplication theorem: P(A ∩ B) = P(A)P(B|A); for independent events P(A ∩ B) = P(A)P(B).

Events are mutually exclusive if P(A ∩ B) = 0 and independent if P(A ∩ B) = P(A)P(B); the two ideas are different.

Total probability: P(A) = Σ P(Eᵢ)P(A|Eᵢ) for a partition E₁, E₂, …, Eₙ of the sample space.

Bayes' theorem: P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ) / Σ P(Eⱼ)P(A|Eⱼ).

Permutations: ⁿPᵣ = n!/(n − r)!; combinations: ⁿCᵣ = n!/[r!(n − r)!]; odds in favour a : b give P = a/(a + b).

A random variable is a real-valued function on the sample space; it is discrete if it takes countable values and continuous otherwise.

For a discrete random variable, the probability mass function satisfies p(x) ≥ 0 and Σp(x) = 1.

For a continuous random variable, the density f(x) satisfies f(x) ≥ 0 and ∫f(x)dx = 1 over its range.

The cumulative distribution function F(x) = P(X ≤ x) is non-decreasing, lies between 0 and 1, with F(−∞) = 0 and F(∞) = 1.

Expectation: E(X) = Σx·p(x) for discrete X and E(X) = ∫x·f(x)dx for continuous X.

E(aX + b) = aE(X) + b, and E(X + Y) = E(X) + E(Y) whether or not X and Y are independent.

Variance: Var(X) = E(X²) − [E(X)]²; Var(aX + b) = a²Var(X).

For independent X and Y, Var(X ± Y) = Var(X) + Var(Y) and E(XY) = E(X)E(Y).

A Bernoulli trial has only two outcomes; the mean of a binomial distribution is np and its variance is npq.

A population is the whole set of units under study; a sample is a subset selected to represent it, and a statistic is computed from the sample.

Simple random sampling gives every unit an equal chance of selection; it may be with or without replacement.

Stratified sampling divides the population into homogeneous strata and samples each; proportional allocation takes nᵢ = n(Nᵢ/N).

Systematic sampling selects every kth unit after a random start, with k = N/n; it is simple but can be biased by periodic data.

Cluster sampling selects whole groups (clusters) rather than individuals; it is cheap but usually less precise.

Sampling error arises because only part of the population is observed; non-sampling errors arise from faulty frames, non-response or recording.

Standard error of the mean: SE = σ/√n, or s/√n when σ is unknown; it decreases as n increases.

A sampling distribution is the probability distribution of a statistic over all possible samples of a given size.

For large n the sampling distribution of x̄ is approximately normal with mean μ and variance σ²/n (central limit theorem).

An index number measures the relative change of a variable or group of variables over time or place, expressed as a percentage with a base period equal to 100.

Simple price relative = (p₁/p₀) × 100; the simple average of price relatives is the unweighted mean of these relatives.

Laspeyres price index: P₀₁ = (Σp₁q₀/Σp₀q₀) × 100; it uses base-year quantities and tends to overstate price rises.

Paasche price index: P₀₁ = (Σp₁q₁/Σp₀q₁) × 100; it uses current-year quantities and tends to understate price rises.

Fisher's ideal index is the geometric mean of Laspeyres and Paasche: P₀₁ = √[(Σp₁q₀/Σp₀q₀)(Σp₁q₁/Σp₀q₁)] × 100; it satisfies both time-reversal and factor-reversal tests.

Marshall–Edgeworth index uses the mean of base and current quantities as weights: Σp₁(q₀ + q₁)/Σp₀(q₀ + q₁).

Time reversal test: P₀₁ × P₁₀ = 1; factor reversal test: P₀₁ × Q₀₁ = Σp₁q₁/Σp₀q₀.

The consumer price index measures the change in the cost of a fixed basket; real wage = (money wage / CPI) × 100.

Chain index numbers link successive relatives, and splicing joins an old series to a new one by multiplying by the ratio at the overlap period.

Statistics has two senses: statistical data (numerical facts) and statistical methods (collection, presentation, analysis, interpretation).

Stages of a statistical study: collection, organisation, presentation, analysis, interpretation - interpretation without analysis is meaningless.

Limitations: statistics studies aggregates of facts, not isolated individuals; results are true only on average and only under comparable conditions.

Primary data are collected by the investigator first-hand; secondary data are already collected by someone else and reused.

Census method: every unit of the population is enumerated; sample method: a part of the population is studied.

Direct personal interview gives the most accurate data but is costly; mailed questionnaires are cheap but suffer low and biased response.

Classification arranges raw data into homogeneous groups; variables are quantitative (discrete or continuous), attributes are qualitative.

A frequency distribution lists classes with frequencies; class mark = (lower + upper limit)/2 and class width = upper − lower limit.

Arithmetic mean (raw data): x̄ = Σx/n; for a frequency distribution x̄ = Σfx/Σf, using class marks for grouped data.

Step-deviation form: x̄ = A + h(Σfd/Σf) with d = (x − A)/h, A an assumed mean and h the class width.

Combined mean: x̄ = (n₁x̄₁ + n₂x̄₂)/(n₁ + n₂).

Median (raw data): arrange in order; odd n gives the ((n+1)/2)th value, even n gives the mean of the (n/2)th and (n/2+1)th values.

Median (grouped data): M = L + [(N/2 − c.f.)/f] × h, with L the lower limit of the median class, c.f. the cumulative frequency before it.

Mode (grouped data): Z = L + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h, f₁ the modal frequency, f₀ and f₂ its neighbours.

Empirical relation: Mode = 3 Median − 2 Mean, valid for moderately skewed distributions.

Geometric mean GM = (x₁x₂…xₙ)^(1/n) suits ratios and growth rates; harmonic mean HM = n/Σ(1/x) suits average speeds.

Mean uses every observation and is pulled by extreme values; median and mode are positional and resistant to outliers.

Range = L − S; coefficient of range = (L − S)/(L + S); range uses only two observations.

Quartile deviation QD = (Q₃ − Q₁)/2; coefficient of QD = (Q₃ − Q₁)/(Q₃ + Q₁).

Mean deviation about A: MD = Σf|x − A|/Σf; it is least when A is the median.

Variance of raw data: σ² = Σx²/n − x̄²; for a frequency distribution σ² = Σfx²/Σf − x̄².

Standard deviation is the positive square root of variance and carries the units of the data.

Coefficient of variation CV = (σ/x̄) × 100 compares variability across units; the smaller CV marks the more consistent series.

Combined variance: σ² = [n₁(σ₁² + d₁²) + n₂(σ₂² + d₂²)]/(n₁ + n₂), with d₁ = x̄₁ − x̄ and d₂ = x̄₂ − x̄.

Adding a constant to every observation leaves variance unchanged; multiplying every observation by k multiplies variance by k².

Dispersion measures are never negative and equal zero only when all observations are identical.

Skewness measures asymmetry; kurtosis measures peakedness or flatness relative to the normal curve.

Pearson's coefficient: SK = (Mean − Mode)/σ, or SK = 3(Mean − Median)/σ when the mode is ill-defined.

Bowley's coefficient: SK = (Q₃ + Q₁ − 2Q₂)/(Q₃ − Q₁), which lies between −1 and +1.

Positive skew (right tail): Mean > Median > Mode; negative skew (left tail): Mean < Median < Mode.

A symmetric distribution has SK = 0 and mean = median = mode.

Moment about the mean: μᵣ = Σf(x − x̄)ʳ/Σf; μ₁ = 0 and μ₂ = σ².

β₁ = μ₃²/μ₂³ and γ₁ = ±√β₁ measure the direction and degree of skewness.

β₂ = μ₄/μ₂² and excess kurtosis γ₂ = β₂ − 3; the mesokurtic normal curve has β₂ = 3 and γ₂ = 0.

β₂ > 3 is leptokurtic (peaked); β₂ < 3 is platykurtic (flat).

Correlation measures the strength and direction of a linear relation between two variables; it does not establish causation.

Karl Pearson's coefficient: r = Σ(x − x̄)(y − ȳ)/√[Σ(x − x̄)² Σ(y − ȳ)²], lying between −1 and +1.

Shortcut form: r = [nΣxy − ΣxΣy]/√{[nΣx² − (Σx)²][nΣy² − (Σy)²]}.

r = +1 is perfect positive, r = −1 perfect negative, r = 0 no linear correlation; r is unaffected by change of origin or scale.

Spearman's rank correlation: ρ = 1 − 6Σd²/[n(n² − 1)], with d the rank difference; repeated ranks need the correction Σ(t³ − t)/12.

Regression of y on x: y − ȳ = b_yx(x − x̄), where b_yx = Σ(x − x̄)(y − ȳ)/Σ(x − x̄)² = r(σ_y/σ_x).

Regression of x on y: x − x̄ = b_xy(y − ȳ), where b_xy = r(σ_x/σ_y); both lines pass through (x̄, ȳ).

r = ±√(b_yx × b_xy) with the sign of r; since |r| ≤ 1, b_yx × b_xy ≤ 1.

If σ_x = σ_y then b_yx = b_xy = r; the regression lines coincide only when r = ±1.

Classical definition: P(A) = favourable outcomes / total equally likely outcomes.

Axioms: 0 ≤ P(A) ≤ 1, P(S) = 1, and P(A ∪ B) = P(A) + P(B) for mutually exclusive A and B.

Addition theorem: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); complement rule: P(A′) = 1 − P(A).

Conditional probability: P(A|B) = P(A ∩ B)/P(B), defined only when P(B) > 0.

Multiplication theorem: P(A ∩ B) = P(A)P(B|A); independent events satisfy P(A ∩ B) = P(A)P(B).

Mutually exclusive means P(A ∩ B) = 0; independent means P(A ∩ B) = P(A)P(B) - the two are distinct ideas.

Total probability: P(A) = Σ P(Eᵢ)P(A|Eᵢ) for a partition E₁, …, Eₙ of the sample space.

Bayes' theorem: P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ)/Σ P(Eⱼ)P(A|Eⱼ).

ⁿPᵣ = n!/(n − r)! and ⁿCᵣ = n!/[r!(n − r)!]; odds a : b in favour give P = a/(a + b).

A random variable is a real-valued function on the sample space; discrete if it takes countably many values, continuous otherwise.

Probability mass function: p(x) ≥ 0 and Σp(x) = 1; probability density: f(x) ≥ 0 and ∫f(x)dx = 1 over the range.

Distribution function F(x) = P(X ≤ x) is non-decreasing with 0 ≤ F(x) ≤ 1, F(−∞) = 0 and F(∞) = 1.

Expectation: E(X) = Σx·p(x) for discrete X and E(X) = ∫x·f(x)dx for continuous X.

E(aX + b) = aE(X) + b, and E(X + Y) = E(X) + E(Y) whether or not X and Y are independent.

Variance: Var(X) = E(X²) − [E(X)]²; Var(aX + b) = a²Var(X).

For independent X and Y: Var(X ± Y) = Var(X) + Var(Y) and E(XY) = E(X)E(Y).

A Bernoulli trial has two outcomes; the binomial distribution has mean np and variance npq.

A population is the whole set of units; a sample is a subset; a statistic is computed from the sample.

Simple random sampling gives every unit an equal chance; it may be with or without replacement.

Stratified sampling splits the population into homogeneous strata; proportional allocation takes nᵢ = n(Nᵢ/N).

Systematic sampling picks every kth unit after a random start, with k = N/n; periodic data can bias it.

Cluster sampling selects whole groups rather than individuals; it is economical but usually less precise.

Sampling error comes from observing only part of the population; non-sampling errors come from faulty frames, non-response or recording.

Standard error of the mean: SE = σ/√n, or s/√n when σ is unknown; it falls as n grows.

A sampling distribution is the probability distribution of a statistic over all samples of a given size.

For large n the sampling distribution of x̄ is approximately normal with mean μ and variance σ²/n (central limit theorem).

An index number expresses relative change over time or place as a percentage, with the base period set to 100.

Simple price relative = (p₁/p₀) × 100; the simple average of price relatives is the unweighted mean of these relatives.

Laspeyres: P₀₁ = (Σp₁q₀/Σp₀q₀) × 100, using base-year quantities; it tends to overstate price rises.

Paasche: P₀₁ = (Σp₁q₁/Σp₀q₁) × 100, using current-year quantities; it tends to understate price rises.

Fisher's ideal index: √[(Σp₁q₀/Σp₀q₀)(Σp₁q₁/Σp₀q₁)] × 100; it satisfies both time-reversal and factor-reversal tests.

Marshall–Edgeworth index uses averaged quantities as weights: Σp₁(q₀ + q₁)/Σp₀(q₀ + q₁).

Time reversal test: P₀₁ × P₁₀ = 1; factor reversal test: P₀₁ × Q₀₁ = Σp₁q₁/Σp₀q₀.

The consumer price index tracks a fixed basket; real wage = (money wage/CPI) × 100.

Chain indices link successive relatives; splicing joins an old series to a new one using the ratio at the overlap period.